Positive real numbers $a,b,$ and $c$ have the property that $\sqrt{\log a}+\sqrt{\log b}+\sqrt{\log c}+\log \sqrt{a}+\log \sqrt{b}+\log \sqrt{c}=148$ and all six terms on the left are positive integers, where $\log$ denotes the base-$10$ logarithm. What is $abc$?